Matrix multiplication
Matrices as transformations, and multiplication as doing one transformation after another.
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Matrix multiplication
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What you'll learn
- A matrix is a transformation written down — its columns are where the basis vectors land.
- Multiplying matrices composes transformations, which is why the order cannot be swapped.
- The row-times-column rule is a consequence of composition, not the definition worth remembering.
From a rule to a reason
Most people meet matrix multiplication as a rule about rows and columns, which is memorable and explains nothing. The geometric reading — a matrix is a transformation of space, and a product is one transformation followed by another — turns the rule into something you can predict rather than recall.
What to watch for
Watch the moment the video derives the arithmetic from the geometry. Once you have seen that the entry at row i, column j is forced by where the basis vectors go, the rule stops being arbitrary. The non-commutativity falls out of the same picture for free.
Where this sits
This is the last node before the network itself. A forward pass is a stack of these transformations with a nonlinearity wedged between each pair, and the reason the nonlinearity has to be there is that without it the whole stack collapses back into one matrix.
Check yourself
Why is AB generally not BA?
Because composition is ordered. Rotating then stretching lands somewhere different from stretching then rotating, and the matrices record exactly that.
What do the columns of a matrix tell you at a glance?
Where each basis vector ends up after the transformation. Reading the columns is usually faster than computing anything.
Go deeper
The maths underneath
Notes
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